Stable phenomena for some automorphism groups in topology Erik Lindell ©Erik Lindell, Stockholm, 2021 Address: Erik Lindell, Matematiska institutionen, Stockholms universitet, 106 91 Stockholm E-mail address: [email protected]

ISBN 978-91-7797-991-3

Printed by Eprint AB 2021 Distributor: Department of Mathematics, Stockholm University Abstract

This licentiate thesis consists of two papers about topics related to representation stability for different automorphisms groups of topological spaces and manifolds. In Paper I, we study the rational homology groups of Torelli groups of smooth, compact and orientable surfaces. The Torelli group of a smooth surface is the group of isotopy classes of orientation preserving diffeomor- phisms that act trivially on the first homology group of the surface. In the paper, we study a certain class of stable homology classes, i.e. classes that exist for sufficiently large genus, and explicitly describe the image of these classes under a higher degree version of the Johnson homomorphism, as a representation of the symplectic group. This gives a lower bound on the dimension of the stable homology of the group, as well as providing some further evidence that these homology groups satisfy representation stability for symplectic groups, in the sense of Church and Farb. In Paper II, we study pointed homotopy automorphisms of iterated wedge sums of spaces as well as boundary relative homotopy automor- phisms of iterated connected sums of manifolds with a disk removed. We prove that the rational homotopy groups of these, for simply connected CW-complexes and closed manifolds respectively, satisfy representation stability for symmetric groups, in the sense of Church and Farb.

3 4 Sammanfattning

Denna licentiatavhandling best˚arav tv˚aartiklar, som b˚adabehand- lar ¨amnenrelaterade till representationstabilitet f¨orautomorfigrupper av olika topologiska rum och m˚angfalder. I Artikel I studeras de rationella homologigrupperna av Torelligrup- per av sl¨ata,kompakta och orienterbara ytor. Torelligruppen av en sl¨at yta ¨argruppen av isotopiklasser av orienteringsbevarande diffeomorfier som verkar trivialt p˚aytans f¨orstahomologigrupp. I artikeln studeras en specifik klass av stabila homologiklasser, dvs. klasser som existerar d˚a ytans genus ¨artillr¨ackligt stort, och bilden av dessa klasser under en vari- ant av Johnsonhomomorfismen i h¨ogrehomologisk grad beskrivs explicit, som en representation av den symplektiska gruppen. Detta ger en undre begr¨ansningp˚agruppens stabila homologi och ¨aven vidare bel¨aggsom pekar p˚aatt dessa homologigrupper uppfyller representationsstabilitet, i den mening som definierats av Church och Farb. I Artikel II studeras baspunktsbevarande homotopiautomorfier av iter- erade kilsummor av rum och randkomponentsrelativa homotopiautomor- fier av sammanh¨angandesummor av m˚angfaldermed en disk borttagen. De rationella homotopigrupperna av dessa, f¨orenkelt sammanh¨angande CW-komplex, respektive slutna m˚angfalder,bevisas uppfylla representa- tionsstabilitet f¨orsymmetriska grupper, i den mening som definierats av Church och Farb.

5 6 Acknowledgments

I would like to express my gratitude to my advisor Dan Petersen for all of his help and general support. His advice and inspiration has been invaluable. My co-advisor, Alexander Berglund, has also given me a lot of inspi- ration, as well as thoughtful advice and feedback, for which I am truly thankful. I would also like to thank Bashar Saleh for inviting me to work on the project that led to the second paper included in this thesis.

7 8 List of papers

The following papers, referred to in the text by their Roman numerals, are included in this thesis.

Paper I: Abelian cycles in the homology of the Torelli group Erik Lindell, Submitted.

Paper II: Representation stability for homotopy automorphisms Erik Lindell, Bashar Saleh

9 10 Contents

Abstract 3

Sammanfattning 5

Acknowledgments 7

List of papers 9

General Introduction 13 1 Introduction ...... 15 2 Torelli groups of surfaces ...... 16 2.1 Mapping class groups ...... 16 2.2 Torelli groups ...... 20 2.3 The homology of Torelli groups ...... 22 2.4 The Johnson homomorphism ...... 23 2.5 Summary of Paper I ...... 24 3 Representation stability and homotopy automorphisms . . 26 3.1 Homological stability ...... 26 3.2 Representation stability ...... 29 3.3 Summary of Paper II ...... 33

Bibliography 37

11 12 General introduction

1. Introduction

This thesis consists of two papers that are not directly related, so let us start by putting these in a common context. The first theme relating the two papers is that they concern questions that arise from studying automorphisms of topological spaces or manifolds, in an appropriate cat- egory. There are several categories that are natural to consider in this context. In the category of topological spaces, the automorphisms of a space X are the self-homeomorphisms, which we denote by Homeo(X). In the category of smooth manifolds, the automorphisms are the self-diffeomorphisms, which we will denote by Diff(X). In homotopy theory, it is more natural to consider homotopy automorphisms than homeomorphisms, i.e. homo- topy self-equivalences. These are precisely those maps that become auto- morphisms in the homotopy category of spaces1 We will use hAut(X) to denote the homotopy automorphisms of a space X, which is a (grouplike) topological monoid, rather than a group. If the type of automorphisms we consider is not specified, we will simply write Aut(X).

Remark 1.1. Since the spaces of automorphisms that we consider are topological groups or grouplike monoids, we may define classifying spaces B Aut(X). The classifying space B hAut(X) classifies fibrations E → B with fiber homotopy equivalent to X. Similarly, B Homeo(X) and B Diff(X) classify topological and smooth fiber bundles E → B with fiber X, respectively.

The second theme connecting the two papers is stability. In each case, we study some sequence of spaces

X1 → X2 → · · · → Xn → · · · that naturally induces maps

Aut(X1) → Aut(X2) → · · · → Aut(Xn) allowing us to ask what happens with the automorphisms when n tends to infinity. More specifically, we study the stabilization behavior of some associated algebraic invariants, such as rational homotopy or homology

1The “correct” notion of the homotopy category uses weak homotopy equiva- lences, rather than homotopy equivalences. In all situations we will consider, the spaces have the homotopy type of CW-complexes, however, so by the Whitehead theorem, we do not have to make this distinction.

15 groups. For this reason, all vector spaces, homology and cohomology will be over the rational numbers, unless otherwise specified. The common context we have now described is much broader than the specific subject matter of Papers I and II. In fact, this general introduc- tion will generally be more meandering than what is strictly necessary to understand the results of these papers. The aim of this is to provide a somewhat broader view into the areas that these papers are part of. In particular, we will review a large number of examples that can be safely skipped by the reader who only wants the background necessary for un- derstanding the specific results of the papers.

2. Torelli groups of surfaces

2.1. Mapping class groups. In many situations, we are only interested in automorphisms of a space up to homotopy equivalence. For an ori- + entable manifold M, we define its mapping class group to be π0 Homeo (M), i.e. the group of isotopy classes of orientation preserving homeomorphisms of M. In Paper I, we study Torelli groups of compact orientable surfaces, which are important subgroups of their mapping class groups and which we will define in the next subsection. For this reason, we will restrict our attention to compact surfaces for the rest of this section.

Remark 2.1. For smooth manifolds, we generally define the mapping class group as the group of isotopy classes of self-diffeomorphisms, rather than homeomorphisms. However, it is a classic result that every home- omorphism of a smooth and compact surface is homotopic to a diffeo- morphism. In fact, in this situation it is also true that every homotopy automorphism is homotopic to a homeomorphism, so for a smooth and compact surface X, we have

∼ ∼ π0 hAut(X) = π0 Homeo(X) = π0 Diff(X).

This means that in the context of Paper I, it is not really important which category we consider. For concreteness, however, we will work primarily with diffeomorphisms.

For g ≥ 0, let Sg denote a closed, orientable surface of genus g, as in + Figure 1. We let Γg := π0 Diff (Sg) denote the mapping class groups of isotopy classes of orientation preserving diffeomorphisms of Sg. We can also consider surfaces with marked points and boundary compo- s nents, which we write as Sg,r if there are r marked points and s boundary

16 components. In these cases, we only consider isotopy classes of diffeomor- phisms that fix the set of marked points and fix the boundary components s pointwise. We write Γg,r for the mapping class group of a surface with r marked points and s boundary components. If there are no marked points or boundary components, we simply omit the corresponding index. We s may note that we have a natural homomorphism Γg,r → Γg, induced by + s + the map Diff (Sg,r) → Diff (Sg) given by forgetting the marked points, gluing disks to the boundary components and extending diffeomorphisms by the identity.

Figure 1: A closed, orientable surface of genus 6.

Remark 2.2. We may also view mapping class groups from a more geo- s metric angle. If we let Mg,r denote the moduli space of complex, smooth and projective curves of genus g with r marked points and s marked non- zero tangent vectors (or, equivalently up to homotopy, Riemann surfaces of genus g with r marked points and s boundary components), it turns out s s that Γg,r is precisely the (orbifold) fundamental group of Mg,r. Moreover, the universal covering orbifold, which is known as Teichm¨uller space, is s s contractible, so Mg,r is in fact a classifying space of Γg,r. From this per- s spective, it can be noted that the map Γg,r → Γg is the map induced on s fundamental groups by the map Mg,r → Mg given by forgetting about the marked points and tangent vectors. The mapping class groups of several surfaces of low genus can be de- scribed explicitly. To get some feeling for the subject, let us take a look at what is known in some of the cases when g = 0, g = 1 and g = 2. As indicated in the introduction, these examples are included to give a broader background and may be safely skipped for a reader eager to get to the results of Paper I as soon as possible. The reader hungry for more examples should instead see for example [FM12], from which the first two of the following examples are taken. Example 2.3. In genus 0, it is easiest to start with the mapping class group Γ0,1. This is because we can identify S0,1 with the one-point com- 2 2 pactification of R and any diffeomorphism of R extends to a diffeomor- phism of S0,1 that fixes the marked point. However, every orientation

17 2 preserving diffeomorphism of R is homotopic to the identity, by simply taking the straight line homotopy, so Γ0,1 is trivial. It follows from this that Γ0 is also trivial, since every diffeomorphism of the sphere is homo- topic to one that fixes a point. For n = 2 and n = 3, it may be proven that ∼ Γ0,n = Σn, the symmetric group on n letters and for n ≥ 4, the mapping class group is the spherical braid group, i.e. the fundamental group of the unordered configuration space of n points on S2.

Example 2.4. Now, let us move on to g ≥ 1. We can note that since + ∼ 2g Diff (Sg) acts on Sg, for any g, we get an action by Γg on H1(Sg, Z) = Z . This induces a homomorphism Γg → GL(2g, Z). We shall see in Section 2.2 that the image of this map actually lies in SL(2g, Z). In the g = 1 case we thus get a homomorphism

σ :Γ1 → SL(2, Z), which is in fact an isomorphism. A complete proof of this can be found, for example, in [FM12], but let us outline the idea. To prove that σ is surjective, note that any element T of SL(2, Z) induces a homeomorphism 2 2 R → R , which is equivariant with respect to the translation action of 2 2 2 ∼ Z . This gives us an induced homeomorphism φT of R /Z = S1 such that σ([φT ]) = T . To prove injectivity, we use that every element of Γ1 has 2 a representative that fixes a point of S1. Furthermore, S1 is a K(Z , 1)- space, so there is a bijective correspondence between homotopy classes of 2 based self-homeomorphisms of S1 and self-homomorphisms of Z . ∼ ∼ From this, it may be deduced that Γ1,1 = Γ1 = SL(2, Z), which in the geometric picture reflects the fact that M1,1 is the moduli space of elliptic curves.

Example 2.5. Already in genus 2 the description of Γg is not as simple. In 2000, it was proven by Bigelow and Budney that there exists a faithful representation Γ2 ,→ GL(64, C), so that Γ2 is in fact a linear group [BB01]. In fact, this result was deduced by first constructing, for every n ≥ 0, a faithful representation

 n − 1  K :Γ ,→ GL n , . n 0,n 2 C

This was based on the earlier results by Bigelow [Big01] and Krammer [Kra00] [Kra02], who showed that the classical braid group Brn has a faithful representation, known as the Lawrence-Krammer representation, for every n ≥ 1. For a general manifold M, we define its n-stranded braid group to be the fundamental group of the unordered configuration space

18 Bn(M) of n points in M. The classical braid group is the braid group of 2 2 ∼ 1 D , a 2-dimensional disk. In this case, we have π1(Bn(D )) = Γ0,n, i.e the mapping class group of a disk with n marked points. 2 If {p1, p2, . . . , pn} are the marked points of S , we have a homomor- phism Brn−1 → Stab(pn), where Stab(pn) is the stabilizer of pn under the 1 action of Γ0,n, induced by the map Γ0,n−1 → Γ0,n, which is given by gluing a disk with a marked point to the boundary component and extending diffeomorphisms by the identity. The map Brn−1 → Stab(pn) is surjective and its kernel is isomorphic to the center of Brn−1, which is just Z. We thus have a central extension

0 → Z → Brn−1 → Stab(pn) → 0.

Bigelow and Budney then used the Larwrence-Krammer representa- tion to construct a non-faithful representation of Brn−1 whose kernel is precisely its center, and thereby obtaining a faithful representation of Stab(pn). This is a subgroup of Γ0,n of finite index, which means that the faithful representation may be extended to the entire group, producing the representation Kn. + They then used that for any g, Diff (Sg) has a special class σ called the hyperelliptic involution, which is defined as rotation by π around the axis illustrated in Figure 2, and which satisfies σ2 = id. This defines an action by Z/2Z on Sg, and the quotient of Sg by this action is a surface of genus 0. The quotient map is a branched cover with 2g + 2 branch points, also illustrated in Figure 2.

Figure 2: The hyperelliptic involution of Sg rotates the surface by π around the dashed axis. The branch points of the corresponding quotient map are marked in red.

This means that any diffeomorphism of Sg that commutes with σ in- duces a diffeomorphism of the sphere, which fixes 2g +2 marked points. If Z/2Z we let Γg denote the subgroup of Γ2 generated by the mapping classes of those diffeomorphisms that commute with σ, this gives us a homomor- Z/2Z phism Γg → Γ0,2g+2, with the kernel isomorphic to Z/2Z. However, it Z/2Z is a classic result that for g = 2 the inclusion map Γ2 → Γ2 is in fact

19 an isomorphism, so we get a representation Γ2 → GL(60, C), with kernel isomorphic to Z/2Z. However, in the representation Γ2 → GL(4, Z), in- troduced in Example 2.4, the mapping class of the hyperelliptic involution is mapped to −1, so if we tensor with this representation we get a faithful representation Γ2 ,→ GL(64, C).

Now let us return to the more general picture and the theory we need to understand the results of Paper I. It is a classical result, independently proven by Dehn and Lickorish, that for each g ≥ 0, the group Γg is finitely generated and a finite generating set is given by Dehn twists around simple, closed curves (see [FM12, Theorem 4.1] for a proof). We can describe a representative of such a mapping class by choosing a tubular neighborhood N of the curve in question. If we identify N with S1 ×[0, 1], the Dehn twist is given by (s, t) 7→ (se2πti, t) on N and by the identity outside of N. We may illustrate how the Dehn twist acts on N as in Figure 3. The isotopy class of a Dehn twist is independent of the choice of tubular neighborhood and Dehn twists around homotopic curves represent the same mapping class. If we are given the homotopy s class of a curve γ in Sg,r, it is thus well-defined to talk about the Dehn twist around γ.

Figure 3: A Dehn twist.

2.2. Torelli groups. From now, let us restrict our interest to surfaces of genus g ≥ 1. As we saw in Example 2.4, the diffeomorphism group + Diff (Sg) acts on the surface Sg, which induces an action by Γg on the first homology group H1(Sg, Z) of the surface, which we will denote by HZ for brevity. As this is the middle homology group of an even-dimensional closed manifold, it is equipped with an intersection form ω, making it into a symplectic abelian group. The homology group is free of rank 2g and a symplectic basis {a1, b1, . . . , ag, bg} is given by the loops illustrated in Figure 4. This means that in this basis, ω can be explicitly characterized by

ω(ai, bj) = −ω(bj, ai) = δij,

ω(ai, aj) = ω(bi, bj) = 0,

20 for all 1 ≤ i, j ≤ g. Any orientation preserving diffeomorphism preserves

ω, which means that we get a group homomorphism from Γg to Sp(HZ), the symplectic group of HZ. This symplectic representation is well un- derstood and can be used to get a lot of information about Γg. However, its kernel is highly non-trivial, and thus important to study, in order to understand the more “mysterious” part of Γg that cannot be understood through the symplectic representation. This kernel is called the Torelli group of Sg, and we will denote it by Ig.

Figure 4: A genus 6 surface with a choice of curves representing a symplectic basis of HZ.

s By composition, we also get maps Γg,r → Sp(HZ) for every r, s ≥ 0 and s we define the Torelli groups Ig,r as the kernels of these. s ∼ Remark 2.6. In the geometric picture, the homomorphism Γg,r → Sp(HZ) = Sp(2g, Z) can be viewed as a reflection of the fact that we have a map s Mg,r → Ag (factoring through Mg), where Ag is the moduli space of principally polarized abelian varieties of dimension g, given by sending a curve to its Jacobian. The (orbifold) fundamental group of Ag is pre- s cisely Sp(2g, Z) and the map Γg,r → Sp(2g, Z) is the map induced on s fundamental groups by Mg,r → Ag. This means that, in some sense, the Torelli group of a surface measures the difference between curves and abelian varieties, which is why it is interesting for algebraic geometers.

Before discussing Torelli groups generally, let us look at some of what is known about them in low genus.

Example 2.7. Let us start with the case g = 1. Since we have already ∼ seen that Γ1 and Γ1,1 are both isomorphic to SL(2, Z) = Sp(2, Z), we get ∼ ∼ that the map to Sp(2, Z) is an isomorphism and thus I1 = I1,1 = 1, the trivial group. In the geometric picture, this is reflective of the fact that elliptic curves are the same things as abelian varieties of dimension 1.

Example 2.8. Already in genus 2, the picture becomes more compli- cated. Unlike Γ2, which we know is generated by a finite number of Dehn twists, I2 is not finitely generated. In fact, it was proven by Mess [Mes92]

21 that I2 is a free group generated by infinitely many separating twists, i.e. Dehn twists around curves that separate the surface into two connected components.

Now let us return to the more general setting. As for Γg, we may describe a generating set for Ig using Dehn twists. We have already in the previous example seen examples of Dehn twists that lie in Ig, that is the separating twists, but these are not enough to generate Ig for higher g. However, we may compose Dehn twists around several curves to get something that works. A bounding pair in Sg is a pair of simple, closed, non-separating and homologous curves. Up to homeomorphism, a bounding pair in Sg is always of the form illustrated in Figure 5. If we Dehn twist along both curves in a bounding pair, but in opposite directions, we get a mapping class in Ig. We call such a mapping class a bounding pair map. It was proven by Johnson that for g ≥ 3, Ig is generated by a finite number of bounding pair maps [Joh83]. Bounding pair maps are one of the main tools used in Paper I.

Figure 5: A typical example of a bounding pair.

2.3. The homology of Torelli groups. The topic of Paper I is the rational homology of Torelli groups. It is a classic result that the map

Γg → Sp(HZ) is surjective, so we get a short exact sequence

1 → Ig → Γg → Sp(HZ) → 1.

This gives rise to a group homomorphism from Sp(HZ) to Out(Ig), the outer automorphism group of Ig, given by choosing a preimage and then taking the outer automorphism class of the conjugation by this preimage. This is well-defined, since two preimages of the same element differ by an element of Ig and thus the corresponding automorphisms of Ig differ by an inner automorphism. Since inner automorphisms induce trivial actions on homology, this gives rise to an Sp(HZ)-action on H∗(Ig). When considering surfaces with one boundary component, which is 1 1 what we do in Paper I, there is a natural map Sg → Sg+1 given by

22 1 2 Figure 6: Gluing an S3 to an S1 along the dashed boundary components to 1 get an S4 . gluing a surface of genus 1 with two boundary components along the 1 boundary component of Sg , as in Figure 6. This induces a homomor- 1 1 phism Ig → Ig+1, given by extending by the identity on the attached component. This gives us a sequence of group homomorphisms

1 1 1 I1 → I2 → · · · → Ig → · · ·

It thus makes sense to ask about the stabilization behavior of this se- quence, as the genus g tends to infinity. In Paper I, we study the induced sequence in rational homology, as a sequence of Sp(2g, Z)-representations. 1 The first rational homology group H1(Ig ) is known for all g ≥ 1. For g = 1 we know that the group is trivial, for g = 2 we have a full description of the group Ig due to Mess (see Example 2.8) which makes the first homology easy to describe, and for g ≥ 3, a complete description was given by Johnson, as we shall see in the next section. In homological degrees at least two, however, quite little in known even 1 about the stable homology of Ig . Unstably, we have already seen that 1 the first rational homology H1(I2 ) cannot be finite-dimensional and it was recently proven by Gaifullin [Gai18] that for g ≥ 3 and 2g − 3 ≤ n ≤ 3g − 6, Hn(Ig, Z) contains a free abelian subgroup of infinite rank, so unstably, the rational homology is not finite dimensional in any degree. It is unknown what happens with the dimension in the stable range.

2.4. The Johnson homomorphism. The main tool in Paper I is the 1 V3 Johnson homomorphism, a group homomorphism Ig → HZ that was ∼ introduced by Dennis Johnson in the early 1980’s. Let H := H1(Sg) = HZ⊗Q. For n ≥ 1, the Johnson homomorphism induces Sp(HZ)-equivariant maps 3 ! n 3 ! 1 ^ ∼ ^ ^ ψn : Hn(Ig ) → Hn HZ = H .

For n = 1, Johnson proved that this map is an isomorphism for g ≥ 3, but already for n = 2, it is known that this map is not surjective.

23 The image was completely described for n = 2 by Hain [Hai97] and for n = 3 by Sakasai [Sak05], up to a single irreducible subrepresentation, which was later independently determined to lie in the image by Kupers and Randal-Williams [KR20]. It remains an open problem whether ψn is stably injective. 1 We can note that for g ≥ 3, H1(Ig ) is a finite dimensional and algebraic representation of the arithmetic group Sp(2g, Z), which means that it is simply a restriction of a representation of Sp(2g, Q). If ψn is stably injective, it is clear that the homology is finite dimensional and algebraic in every degree. Conversely, it was proven by Kupers and Randal-Williams 1 that if Hn(Ig ) is stably finite dimensional and algebraic, then ψn is stably injective and it is possible to give an explicit description of its cokernel [KR20].

2.5. Summary of Paper I. In Paper I, we give large lower bound on the stable image of ψ , for arbitrary n ≥ 1. In order to state this result, n   let us recall some representation theory. As the codomain Vn V3 H of

ψn is an algebraic representation of the arithmetic group Sp(HZ), we may describe the image of ψn in terms of its decomposition into irreducible representations of Sp(H). These irreducibles are indexed by partitions.A partition is a non-increasing sequence λ = (λ1 ≥ λ2 ≥ · · · λk ≥ 0 ≥ · · · ) of non-negative integers and we write the corresponding irreducible repre- sentation as Vλ. The weight of a partition λ is the sum |λ| := λ1 +λ2 +··· . The tensor power H⊗n of the standard representation decomposes into a   direct sum of irreducibles of weight at most n, so Vn V3 H decomposes into irreducibles of weight at most 3n. The main result of Paper I is that all of these top weight irreducibles are contained in the image of ψn: Theorem I.A. For g ≥ 3n, the image of ψ contains all irreducible   n subrepresentations of Vn V3 H of weight 3n.

In fact, we show a slightly more general result, that detects many ir- reducible representations of lower weight in the image as well, but whose statement is a bit more involved.

1 Remark 2.9. There is a homomorphism Ig → Ig,1, induced by gluing 1 a disk with a marked point to the boundary of Sg and extending dif- feomorphisms by the identity. The Johnson homomorphism factors as 1 V3 Ig → Ig,1 → HZ, so in particular, any lower bound on the image of ψn 0 Vn V3  is also a bound on the corresponding map ψn : Hn(Ig,1) → H . 0 This means that Theorem I.A holds for ψn as well.

24 We prove these results by considering a type of homology classes in 1 Hn(Ig ) called abelian cycles. For a general group G, we can define an abelian cycle in Hn(G) by specifying an n-tuple (g1, ··· , gn) of pairwise n commuting elements of G. Such an n-tuple defines a map Z → G, which n induces a map Hn(Z ) → Hn(G). We define the abelian cycle determined n by (g1, ··· , gn) to be the image of the fundamental class [Z ] under this map. n

Figure 7: The bounding pairs defining the abelian cycle used in the proof of Theorem I.A .

1 In Hn(Ig ) we have a class of abelian cycles which are especially easy to construct, given by taking n-tuples of bounding pair maps determined by pairwise disjoint bounding pairs. It turns out that the map ψn is also especially easy to evaluate on abelian cycles. When we have the explicit   images in Vn V3 H of these abelian cycles, we can use symplectic Schur- Weyl duality to detect top weight irreducible subrepresentations. It turns   out that the subrepresentation of Vn V3 H which is spanned by all top weight irreducibles is actually cyclic, and generated by the image of the abelian cycle determined by the bounding pairs in Figure 7. As the abelian cycle illustrated in Figure 7 indicates, it may be viewed as a power of an 1 abelian cycle of degree 1 in the homology of I3 . By considering similar products of more general abelian cycles on surfaces of lower genus, we may detect lower weight irreducibles as well. In this paper, we also prove a secondary result, about the abelian cycles determined by pairwise disjoint bounding pair maps:

1 1 Theorem I.B. Let n ≥ 2 and An(Ig ) ⊂ Hn(Ig ) be the subrepresentation generated by abelian cycles determined by pairwise disjoint bounding pair 1 maps. Then ψn(An(Ig )) is concentrated in weights n, n + 2,..., 3n.

25 Already for n = 3, we know that the image of ψn contains a subrepre- 1 sentation of weight 1, which indicates that the subrepresentation An(Ig ) is likely proper for every n ≥ 3. This theorem thus describes a likely limitation of the method used in this paper.

3. Representation stability and homotopy automorphisms

In Paper II, we study a phenomenon known as representation stability, which we will see is related to the subject of Paper I, as the name indicates. Representation stability is a development of the more classical notion of homological stability, which we will review first.

3.1. Homological stability. Homological stability was originally de- fined for groups. If R is some commutative ring, we say that a sequence

G1 → G2 → · · · of groups is homologically stable, with coefficients in R, if there, for every k ≥ 0, exists some sufficiently large N such that Hk(Gi,R) → Hk(Gi+1,R) is an isomorphism if i > N. We will generally use rationally coefficients, unless otherwise stated. Note that this is equivalent to saying that the sequence BG1 → BG2 → · · · of classifying spaces stabilize in the same sense. We can therefore gener- alize this definition to sequences of spaces in general. Homological stability as a concept was spearheaded by Quillen in the 1970’s [Qui73]. Since then, homological stability has been proven to hold for many sequences of groups and spaces. Let us review some significant examples. As the examples in Section 2, these are primarily included to give a broader background, and the reader eager to get to the results of Paper II may safely skip to Section 3.2.

Example 3.1. The most classical examples of homological stability are for different linear groups. i.e. subgroups of GL(n, R), for different rings, which have natural stabilization maps. For example, different kinds of ho- mological stability for general linear groups have been proven by Quillen, Dwyer [Dwy80], van der Kallen [Kal80] and Charney [Cha80], for orthog- onal groups by Vogtmann [Vog81], Betley [Bet86] and Charney [Cha87], for symplectic groups by Charney [Cha87], Mirazaii and van der Kallen [MK01a] and for unitary groups by Mirazaii and van der Kallen [MK01b].

26 Example 3.2. From our perspective, perhaps one of the most interesting examples of homological stability is for mapping class groups of surfaces. 1 1 As we have seen, we have stabilization maps Sg → Sg+1 given by gluing 1 1 as in Figure 6, which induce stabilization maps Γg → Γg+1. Homologi- cal stability for this sequence was proven by Harer [Har85]. The stable (co)homology was conjectured by Mumford [Mum83] in 1983 to be of the form ∗ 1 ∼ lim H (Γ , ) = [κ1, κ2,...], ← g Q Q i.e. a polynomial algebra with a countable number of generators, known as κ-classes. It remained until 2007 before this conjecture was verified, by Madsen and Weiss [MW07], which is why this result is now known as the Madsen-Weiss theorem. Madsen and Weiss actually proved the stronger result that there is an integral homology equivalence

hocolim BΓ1 → Ω∞MTSO(2), g→∞ g 0 where MTSO(2) is a certain Thom spectrum, built from Grassmannians n of planes in R , whose cohomology is easily computable. This proof has been incredibly influential and led to many results using similar methods. A notable example is the analogous result for mapping class groups of the higher dimensional analogues of surfaces of genus g, i.e. iterated connected sums #g(Sn × Sn). For n ≥ 3, whose stable cohomology was computed by Galatius and Randal-Williams [GR18]. The case when n = 2 is still open, however. Another notable example which initially might seem quite different, but is very analogous, is for automorphism groups of free groups. If Fn denotes the free group on n generators, we let Aut(Fn) denote its auto- morphism group, in the category of groups. Then we have a natural sta- bilization map Aut(Fn) → Aut(Fn+1) given by extending by the identity on the extra generator. Integral homological stability for this sequence was proven by Hatcher, Vogtmann [HV04] and Wahl [HVW06]. Using methods similar to the proof of the Madsen-Weiss theorem, the stable in- tegral homology was later computed by Galatius [Gal11], and determined to be trivial.

Example 3.3. Another example of homological stablity, which is of di- rect relevance to Paper II, is for homotopy automorphisms of iterated connected sums of manifolds. If we let M be a d-dimensional closed and smooth manifold , we can define

n d Mn := (# M) \ int(D ),

27 the n-connected sum of M with itself, with the interior of an d-dimensional disk Dd removed, to create a manifold with boundary. We then have d d stabilization maps Mn → Mn+1, given by gluing M \ int(D t D ) to the boundary component of Mn, along one of its boundary components. Now let hAut∂(Mn) denote the submonoid of homotopy automorphisms on Mn that fix the boundary. Then we naturally get induced stabilization maps hAut∂(Mn) → hAut∂(Mn+1), by extending by the identity. Homological stability for this sequence was proven by Berglund and Madsen [BM20] in the case when M = Sn × Sn, for n ≥ 3, and later by Grey [Gre19] in the case when M = Sn × Sm, for n, m ≥ 3.

Example 3.4. An example of somewhat different flavor is given by un- ordered configuration spaces of manifolds. If M is a manifold, let us use Cn(M) to denote the ordered configuration space of n points in M, and let Bn(M) := Cn(M)/Σn be the unordered configuration space. The study of homological stability for such sequences has a long history, starting 2 with the proof of integral homological stability for Bn(R ) by Arnol’d 2 [Arn69] in 1969. In this case, by identifying R with C, we may define 2 2 maps Bn(R ) → Bn+1(R ) by mapping

{z1, z2, . . . , zn} 7→ {z1, z2, . . . , zn, 1 + max Re(zi)}, i where Re(zi) denotes the real part of zi. A more general method was found by McDuff [McD75], in the case of open manifolds. If M is the interior of a compact manifolds with boundary, we may choose a neighborhood U of the boundary, such that ∼ ∼ M \ U = M, inducing a homeomorphism Bn(M \ U) = Bn(M). For a given choice of base point in U, we can then define a map Bn(M \ U) → Bn+1(M), by adding this base point. Using this, McDuff was able to prove integral homological stability for general open manifolds. Here she also introduced a method known as scanning, which has since been integral in proving homological stability results. Homological stability for open manifolds was also later proven independently by Segal [Seg79]. For closed manifolds, integral homological stability of Bn(M) does not hold. A counterexample is given by M = S2, in which case it may be proven that 2 ∼ H1(Bn(S ), Z) = Z/(2n − 2)Z, which does not stabilize. As we can see this counterexample does not work rationally. It was not until 2011, however, that it was proven by Church [Chu12] that ra- tional homological stability for Bn(M) holds for all connected manifolds

28 with finite dimensional cohomology. To ger around the problem of find- ing a map Bn(M) → Bn+1(M), Church instead worked with the ordered configuration spaces Cn(M), where we instead have maps Cn+1(M) → k Cn(M), given by forgetting the last point, inducing maps H (Cn(M)) → k H (Cn+1(M)) on cohomology. These maps do not stabilize, however, so at a first glance this may look like a fruitless approach. However, Church’s idea was that instead of considering classical homological sta- bility, we should consider representation stability, a concept that he had introduced together with Farb [CF13] only the year before. We shall in- troduce representation stability in the next section, but the point here is k that Church proved that if we consider the sequence of H (Cn(M)) as a sequence of representations of the symmetric group Σn, it satisfies repre- sentation stability, which in particular proves that the Σn-invariants of k H (Cn(M)) stabilize in the classical sense. These invariants are isomor- k phic to H (Bn(M)), which gives us the desired result.

3.2. Representation stability. As we saw in Example 3.4, there are natural and interesting examples of sequences of spaces where homological stability is not satisfied. In 2010, it was noticed by Church and Farb that there are, however, many cases where the homology groups still stabilize in a weaker sense, when considered as group representations [CF13]. To give a trivial example, consider the sequence of surfaces

1 1 S1 ,→ S2 ,→ · · · , where the inclusions are given by gluing a surface of genus 1 with two 1 boundary components along the boundary component. Then dimQ H1(Sg ) = 2g, so the homology groups certainly do not stabilize in the classical sense. 1 However, H1(Sg ) is the standard representation of Sp(2g, Q) for each g ≥ 1, which corresponds to the partition λ = (1 ≥ 0 ≥ · · · ). This means that in some sense, the homology groups do stabilize, considered as representations of the family {Sp(2g, Q)}g≥1. It is exactly this kind of phenomenon that is captured by representation stability. A more consequential example that we have actually already encoun- tered is for Torelli groups of surfaces. Recall that the sequence of surfaces above induces a sequence of Torelli groups

1 2 Ig → Ig → · · · , which gives rise to a consistent sequence of Sp(2g, Z)-representations in homology. In homological degree 1, we know from the work of Johnson

29 1 ∼ V3 that for g ≥ 3, H1(Ig ) = H1(Sg), which is an algebraic representa- tion, isomorphic to the representation V111 ⊕ V1 of Sp(2g, Q). Thus these homology groups also stabilize, in this representation theoretic sense. Representation stability can be defined for sequences of representa- tions of severel different families of groups, for example Σn, SL(n, Q) and Sp(2n, Q). From now, we let Gn denote any of these groups. The general setup is as follows. Let φ1 φ2 V1 −→ V2 −→ · · · be a sequence, where Vn is a representation of Gn. We say that this sequence is consistent if, for each g ∈ Gn, the square

φn Vn Vn+1 g g

φn Vn Vn+1 is commutative, where we consider g as an element of Gn+1 under the standard inclusion in the right vertical arrow. In order to state the definition of representation stability for all three of the families of groups that we consider, we need a common way to index their irreducible representations. Irreducible representations of SL(n, Q) and Sp(2n, Q) are indexed by partitions. If λ is a partition, let us use V (λ)n to denote either the corresponding irreducible representation of SL(n, Q) or the corresponding irreducible representation of Sp(2n, Q), de- pending on the context. The irreducible representations of Σn are classi- fied by partitions of weight n. For a partition λ = (λ1 ≥ λ2 ≥ · · · ≥ λl) of weight k and an arbitrary integer n ≥ k + λ1, we may define the padded partition λ[n] := (n − k ≥ λ1 ≥ λ2 ≥ · · · ≥ λl), which is of weight n. If Gn = Σn, we let V (λ)n := Vλ[n]. This allows us to make the following definition:

Definition 3.5. Let {Vn, φn}n≥1 be a consistent sequence of represen- tations of Gn. We say that this sequence is representation stable if the following conditions hold, for n sufficiently large:

1. The map φn : Vn → Vn+1 is injective.

2. The span of the Gn+1-orbit of φn(Vn) equals all of Vn+1.

3. If we decompose Vn into irreducible representations as M Vn = cλ,nV (λ)n λ

30 with multiplicities 0 ≤ cλ,n ≤ ∞, then for each λ, the multiplicities cλ,n are eventually independent of n.

We have already seen how representation stability was useful in proving homological stability for unordered configuration spaces. Let us look at this example in a bit more detail, and at some other examples where rep- resentation stability is proven or conjectured. Once again, these examples are not needed to understand the results of Paper II, but included to put them in a broader context.

Example 3.6. We have already seen that Church used representation ∗ stability for H (Cn(M)) to prove homological stability for Bn(M), for 2 connected manifolds M with finite dimensional cohomology. For M = R , representation stability was proven already in [CF13], and this was the initial motivating example for representation stability given by Church and Farb. As we saw in Example 2.5, the fundamental group of Bn(M) is known as the n-stranded braid group on M and similarly, the fundamental group of Cn(M) is known as the n-stranded pure braid group on M. Let us denote these groups by Brn(M) and PBrn(M), respectively. We have a natural homomorphism Brn(M) → Σn, given by mapping the class of a loop to the permutation it induces on the chosen base point. It is clear that this is a surjection and that the pure braid group lies in its kernel, so we get a short exact sequence

1 → PBrn(M) → Brn(M) → Σn → 1.

As we saw in Section 2.3, this means that we get a Σn-action on PBrn(M). In fact, Bn(M) is a classifying space for Brn(M) and Cn(M) is a classify- ing space for PBrn(M), so the group homology of these groups agree with the homology of the corresponding spaces. The Σn-action on H∗(PBrn(M)) agrees with the natural Σn-action on H∗(Cn(M)) induced by the action on Cn(M). Thus it follows by the results we have reviewed that for M con- nected and with finite dimensional cohomology, the corresponding braid groups satisfy homological stability and the pure braid groups satisfy rep- resentation stability.

Example 3.7. Representation stability has mostly been proven for the homology of different groups and spaces. Recently however, it was proven by Kupers and Miller [KM18] that for M simply connected and of dimen- sion at least 3, the rational homotopy groups of Bn(M) are representation stable.

31 Example 3.8. There are several examples similar to the motivating ex- ample of pure braid groups, where representation stability is conjectured to hold. For example, consider the short exact sequence

1 1 1 → Ig → Γg → Sp(2g, Z) → 1 that we saw in Section 2.3. The Sp(2g, Z)-action on the homology of the Torelli grup induced by this sequence makes

1 1 Hk(I1 ) → Hk(I2 ) → · · · into a consistent sequence of Sp(2g, Z)-representations, for every k ≥ 1. For k = 1, we have already seen that representation stability is satisfied. 1 fd 1 Letting Hk(Ig ) denote the subrepresentation of Hk(Ig ) consisting of those vectors whose Sp(2g, Z)-orbits span finite dimensional vector spaces, 1 fd it was conjectured by Church and Farb [CF13] that Hk(Ig ) is stably algebraic and that the corresponding Sp(2g, Q)-representation is repre- sentation stable, for all k ≥ 1. Evidence for this conjecture was provided in their following work [CF12], where they constructed a family of stable 1 classes in Hk(Ig ). Paper I in this thesis provides further similar such evi- dence. It was proven by Boldsen and Dollerup [BH12] that the surjectivity 1 condition, i.e. condition 2, of Definition 3.5 is satisified by H2(Ig ). 1 fd 1 Under the assumption that Hk(Ig ) = Hk(Ig ) stably and that this representation is stably algebraic, representation stability follows from the results by Kupers and Randal-Williams [KR20] described in Section 2.4. Example 3.9. We also have an analogous example for automorphism groups of free groups, where representation stability is conjectured to hold. If Fn is the free group on n letters, its first homology is simply the ∼ n abelianization H1(Fn, Z) = Z . Similarly as for the mapping class group, we thus get a homomorphism Aut(Fn) → GL(n, Z). This homomorphism is surjective, so if we let I Aut(Fn) denote its kernel, we get a short exact sequence 1 → I Aut(Fn) → Aut(Fn) → GL(n, Z) → 1.

In analogy with with mapping class groups, we call the kernel I Aut(Fn) the Torelli subgroup of Aut(Fn). We have natural inclusions I Aut(Fn) → I Aut(Fn+1) given by extending an automorphism by the identity on the last generator, and for each k ≥ 0, this makes

Hk(I Aut(F1)) → Hk(I Aut(F2)) → · · · into a consistent sequence of GL(n, Z)-representations, which is conjec- tured to be representation stable in the same sense as in Example 3.8.

32 Let us return to the general situation. The conditions in Definition 3.5 are quite explicit and therefore often difficult to verify directly. Church, Ellenberg and Farb [CEF15] realized that when Gn = Σn, the data of most consistent sequences appearing naturally can be encoded using something they called an FI-module. An FI-module, over some commutative ring R, is simply a functor FI → ModR, where FI is the category of finite sets and injective maps and ModR is the category of R-modules. We shall only be interested in the case when R = Q and will therefore suppress the R from our notation. If V is an FI-module and n = {1, 2, . . . , n}, we may set Vn = V (n) and let φn : Vn → Vn+1 be the map induced by the canonical inclusion n ,→ n + 1. It is easily verified that this defines a consistent sequence of Σn-representations. What makes this construction useful is that represen- tation stability for consistent sequences of Σn-representations corresponds to a finite generation property for FI-modules. Definition 3.10. A FI-module V is said to be finitely generated if there F exists a finite set S ⊂ n≥1 Vn such that there is no proper sub-FI-module F W of V such that S ⊂ n≥1 Wn. Theorem 3.11 (Church, Ellenberg, Farb). An FI-module V is finitely generated if and only if the sequence {Vn}n≥1 is representation stable and each Vn is finite dimensional. The realization that FI-modules was an appropriate tool for studying representation stability of symmetric groups has lead to a wealth of re- search in this direction and similar categories, whose modules are used to study representation stability of other families of groups, have now been described as well.

3.3. Summary of Paper II. In this paper, we study representation stability of homotopy automorphisms in two different settings. In the first setting, let (X, ∗) be a simple connected space with the homotopy Wn type of a finite CW-complex. We define Xn := i=1 X to be the n-fold wedge sum of X with itself. A homotopy automorphism of Xn, which fixes the base point, can easily be extended by the identity to a homotopy automorphism of Xn+1, which also fixes the basepoint. We thus get the sequence hAut∗(X1) → hAut∗(X2) → · · · →, of base-point preserving homotopy automorphisms, which induces a se- quence

Q Q πk (hAut∗(X1)) → πk (hAut∗(X2)) → · · · (3.1)

33 of rational homotopy groups. There is a Σn-action on Xn, given by per- muting the wedge summands, which induces a Σn-action on hAut(Xn) Q and thereby on πk (hAut(Xn)). This action makes the sequence (3.1) into a consistent sequence of Σn-representations, which defines an FI-module. The first main result of Paper II is that for each k ≥ 1, this FI-module is finitely generated, and thus the consistent sequence is representation stable.

Theorem II.A. For every k ≥ 1, the sequence (3.1) is representation stable.

In the second setting we let M be a d-dimensional manifold, which is closed, simply connected and such that M \ int(Dd) is also simply connected. In this situation, we may similarly define Mn as in Example 3.3 to be the manifold we get by removing a d-dimensional disk from the n-fold connected sum of M with itself. Given a homotopy automorphism of Mn which fixes the boundary, we can easily extend it to Mn+1 by the identity. We thus get a sequence

hAut∂(M1) → hAut∂(M2) → · · · , which once again gives us a sequence

Q Q πk (hAut∂(M1)) → πk (hAut∂(M2)) → · · · (3.2) of rational homotopy groups, for each k ≥ 1. In this case, there is no obvious Σn-action on Mn. However, in Paper II, we prove that there is a Q non-trivial Σn-action on πk (hAut∂(Mn)), which makes the sequence (3.2) into a consistent sequence of Σn-representations. The second main result of Paper II is that the FI-module defined by this consistent sequence is also finitely generated.

Theorem II.B. For every k ≥ 1, the sequence (3.2) is representation stable.

We prove these results using rational homotopy theory. The rational ho- motopy theory of simply connected spaces is, loosely speaking, equivalent to the rational homotopy theory of differential graded Lie algebras. The rational homotopy groups of such a space X may therefore be computed by a differential graded Lie algebra (L, d), which we call a Lie model of X, and which satisfies ∼ Q sH(L, d) = π∗ (X), where s denotes suspension and H denotes homology.

34 For a space X, its space of homotopy automorphisms hAut(X) is gen- erally not simply connected, but if we take the universal cover of its classifying space, which we denote by B hAut(X), we have ∼ ∼ πk(hAut(X)) = πk+1(B hAut(X)) = πk+1(B hAut(X)).

+ −1 A Lie model for B hAut(X) is given by Der (L(s H∗(X))), the Lie algebra of derivations of positive degree on the free Lie algebra generated −1 by s H∗(X), the desuspended homology of X. In the settings described above, where we look at homotopy automorphisms that fix a base point or boundary component, we must instead look at a sub-Lie algebra of this. For Xn and Mn, these derivation Lie algebras naturally form consistent sequences of differential graded Lie algebras that are also representations of Σn and with Σn-equivariant differentials. In Paper II, we show that the homology of these sequences are equal to the sequences 3.1 and 3.2, respectively, and that the homology is a finitely generated FI-module in each degree.

35 36 Bibliography

[Arn69] V. I. Arnol’d. “The cohomology ring of the group of dyed braids”. In: Mat. Zametki 5 (1969), pp. 227–231. issn: 0025- 567X. [BM20] A. Berglund and I. Madsen. “Rational homotopy theory of au- tomorphisms of manifolds”. In: Acta Math. 224 (2020), pp. 67– 185. [Bet86] Stanislaw Betley. “Hyperbolic posets and homology stabil- ity for On,n”. In: Journal of Pure and Applied Algebra 43.1 (1986), pp. 1–9. issn: 0022-4049. doi: https://doi.org/ 10 . 1016 / 0022 - 4049(86 ) 90001 - 0. url: https : / / www . sciencedirect.com/science/article/pii/0022404986900010. [Big01] Stephen J. Bigelow. “Braid groups are linear”. In: J. Amer. Math. Soc. 14.2 (2001), pp. 471–486. issn: 0894-0347. doi: 10.1090/S0894-0347-00-00361-1. url: https://doi.org/ 10.1090/S0894-0347-00-00361-1. [BB01] Stephen J. Bigelow and Ryan D. Budney. “The mapping class group of a genus two surface is linear”. In: Algebr. Geom. Topol. 1 (2001), pp. 699–708. issn: 1472-2747. doi: 10.2140/ agt.2001.1.699. url: https://doi.org/10.2140/agt. 2001.1.699. [BH12] Søren K. Boldsen and Mia Hauge Dollerup. “Towards rep- resentation stability for the second homology of the Torelli group”. In: Geom. Topol. 16.3 (2012), pp. 1725–1765. issn: 1465-3060. doi: 10.2140/gt.2012.16.1725. url: https: //doi.org/10.2140/gt.2012.16.1725. [Cha87] Ruth Charney. “A generalization of a theorem of Vogtmann”. In: Proceedings of the Northwestern conference on cohomology of groups (Evanston, Ill., 1985). Vol. 44. 1-3. 1987, pp. 107– 125. doi: 10.1016/0022- 4049(87)90019- 3. url: https: //doi.org/10.1016/0022-4049(87)90019-3.

37 [Cha80] Ruth M. Charney. “Homology stability for GLn of a Dedekind domain”. In: Invent. Math. 56.1 (1980), pp. 1–17. issn: 0020- 9910. doi: 10.1007/BF01403153. url: https://doi.org/ 10.1007/BF01403153. [CEF15] T. Church, J. S. Ellenberg, and B. Farb. “FI-modules and stability for representations of symmetric groups”. In: Duke Mathematical Journal 164.9 (2015), pp. 1833–1910. doi: 10. 1215/00127094-3120274. [Chu12] Thomas Church. “Homological stability for configuration spaces of manifolds”. In: Invent. Math. 188.2 (2012), pp. 465–504. issn: 0020-9910. doi: 10.1007/s00222- 011- 0353- 4. url: https://doi.org/10.1007/s00222-011-0353-4. [CF12] Thomas Church and Benson Farb. “Parameterized Abel-Jacobi maps and abelian cycles in the Torelli group”. In: J. Topol. 5.1 (2012), pp. 15–38. issn: 1753-8416. doi: 10.1112/jtopol/ jtr026. url: https://doi.org/10.1112/jtopol/jtr026. [CF13] Thomas Church and Benson Farb. “Representation theory and homological stability”. In: Advances in Mathematics 245 (Oct. 2013), pp. 250–314. issn: 0001-8708. doi: 10.1016/j. aim.2013.06.016. url: http://dx.doi.org/10.1016/j. aim.2013.06.016. [Dwy80] W. G. Dwyer. “Twisted homological stability for general linear groups”. In: Ann. of Math. (2) 111.2 (1980), pp. 239–251. issn: 0003-486X. doi: 10.2307/1971200. url: https://doi.org/ 10.2307/1971200. [FM12] Benson Farb and Dan Margalit. A primer on mapping class groups. Vol. 49. Princeton Mathematical Series. Princeton Uni- versity Press, Princeton, NJ, 2012, pp. xiv+472. isbn: 978-0- 691-14794-9. [Gai18] Alexander A. Gaifullin. On infinitely generated homology of Torelli groups. 2018. arXiv: 1803.09311 [math.GR]. [Gal11] Søren Galatius. “Stable homology of automorphism groups of free groups”. In: Ann. of Math. (2) 173.2 (2011), pp. 705–768. issn: 0003-486X. doi: 10.4007/annals.2011.173.2.3. url: https://doi.org/10.4007/annals.2011.173.2.3.

38 [GR18] Søren Galatius and Oscar Randal-Williams. “Homological sta- bility for moduli spaces of high dimensional manifolds. I”. In: J. Amer. Math. Soc. 31.1 (2018), pp. 215–264. issn: 0894- 0347. doi: 10.1090/jams/884. url: https://doi.org/10. 1090/jams/884. [Gre19] M. Grey. “On rational homological stability for block auto- morphisms of connected sums of products of spheres”. In: Al- gebraic & Geometric Topology 19.7 (2019), pp. 3359–3407. [Hai97] Richard Hain. “Infinitesimal presentations of the Torelli groups”. In: J. Amer. Math. Soc. 10.3 (1997), pp. 597–651. issn: 0894- 0347. doi: 10.1090/S0894-0347-97-00235-X. url: https: //doi.org/10.1090/S0894-0347-97-00235-X. [Har85] John L. Harer. “Stability of the Homology of the Mapping Class Groups of Orientable Surfaces”. In: Annals of Mathe- matics 121.2 (1985), pp. 215–249. issn: 0003486X. url: http: //www.jstor.org/stable/1971172. [HVW06] Allan Hatcher, Karen Vogtmann, and Natalie Wahl. “Erra- tum to: “Homology stability for outer automorphism groups of free groups [Algebr. Geom. Topol. 4 (2004), 1253–1272 (elec- tronic); MR2113904] by Hatcher and Vogtmann”. In: Algebr. Geom. Topol. 6 (2006), pp. 573–579. issn: 1472-2747. doi: 10.2140/agt.2006.6.573. url: https://doi.org/10. 2140/agt.2006.6.573. [HV04] Allen Hatcher and Karen Vogtmann. “Homology stability for outer automorphism groups of free groups”. In: Algebr. Geom. Topol. 4 (2004), pp. 1253–1272. issn: 1472-2747. doi: 10 . 2140/agt.2004.4.1253. url: https://doi.org/10.2140/ agt.2004.4.1253. [Joh83] Dennis Johnson. “The structure of the Torelli group. I. A finite set of generators for I”. In: Ann. of Math. (2) 118.3 (1983), pp. 423–442. issn: 0003-486X. doi: 10.2307/2006977. url: https://doi.org/10.2307/2006977. [Kal80] Wilberd van der Kallen. “Homology stability for linear groups”. In: Invent. Math. 60.3 (1980), pp. 269–295. issn: 0020-9910. doi: 10 . 1007 / BF01390018. url: https : / / doi . org / 10 . 1007/BF01390018. [Kra00] Daan Krammer. “The braid group B4 is linear”. In: Inven- tiones mathematicae 142 (Dec. 2000), pp. 451–486. doi: 10. 1007/s002220000088.

39 [Kra02] Daan Krammer. “Braid groups are linear”. In: Ann. of Math. (2) 155.1 (2002), pp. 131–156. issn: 0003-486X. doi: 10.2307/ 3062152. url: https://doi.org/10.2307/3062152. [KM18] Alexander Kupers and Jeremy Miller. “Representation stabil- ity for homotopy groups of configuration spaces”. In: J. Reine Angew. Math. 737 (2018), pp. 217–253. issn: 0075-4102. doi: 10.1515/crelle-2015-0038. url: https://doi.org/10. 1515/crelle-2015-0038. [KR20] Alexander Kupers and Oscar Randal-Williams. “ON THE COHOMOLOGY OF TORELLI GROUPS”. In: Forum Math. Pi 8 (2020), e7, 83. doi: 10.1017/fmp.2020.5. url: https: //doi.org/10.1017/fmp.2020.5. [MW07] Ib Madsen and Michael Weiss. “The stable moduli space of Riemann surfaces: Mumford’s conjecture”. In: Ann. of Math. (2) 165.3 (2007), pp. 843–941. issn: 0003-486X. doi: 10.4007/ annals.2007.165.843. url: https://doi.org/10.4007/ annals.2007.165.843. [McD75] Dusa McDuff. “Configuration spaces of positive and negative particles”. In: Topology 14 (1975), pp. 91–107. issn: 0040-9383. [Mes92] Geoffrey Mess. “The Torelli groups for genus 2 and 3 sur- faces”. In: Topology 31.4 (1992), pp. 775–790. issn: 0040-9383. doi: 10.1016/0040-9383(92)90008-6. url: https://doi. org/10.1016/0040-9383(92)90008-6. [MK01a] B. Mirzaii and W. Kallen. “Homology Stability for Symplectic Groups”. In: (Oct. 2001). [MK01b] Behrooz Mirzaii and Der Kallen. “Homology Stability For Unitary Groups”. In: Documenta Mathematica 7 (Dec. 2001). [Mum83] David Mumford. “Towards an enumerative geometry of the moduli space of curves”. In: Arithmetic and geometry, Vol. II. Vol. 36. Progr. Math. Birkh¨auserBoston, Boston, MA, 1983, pp. 271–328.

[Qui73] Daniel Quillen. “Finite generation of the groups Ki of rings of algebraic integers”. In: Higher K-Theories. Vol. 341. Lec- ture Notes in Mathematics. Springer, Berlin, Heidelberg, 1973, pp. 179–198.

40 [Sak05] Takuya Sakasai. “The Johnson homomorphism and the third rational cohomology group of the Torelli group”. In: Topology Appl. 148.1-3 (2005), pp. 83–111. issn: 0166-8641. doi: 10. 1016/j.topol.2004.08.002. url: https://doi.org/10. 1016/j.topol.2004.08.002. [Seg79] Graeme Segal. “The topology of spaces of rational functions”. In: Acta Math. 143.1-2 (1979), pp. 39–72. issn: 0001-5962. doi: 10 . 1007 / BF02392088. url: https : / / doi . org / 10 . 1007/BF02392088. [Vog81] Karen Vogtmann. “Spherical posets and homology stability for 0n,n”. In: Topology 20.2 (1981), pp. 119–132. issn: 0040- 9383. doi: https : / / doi . org / 10 . 1016 / 0040 - 9383(81 ) 90032-X. url: https://www.sciencedirect.com/science/ article/pii/004093838190032X.

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